A268781 -id:A268781 - cp's OEIS frontend

A143870 Number of ways of placing kings with no more than 1 mutual attack on an n X n chessboard.

1, 2, 11, 91, 1462, 38483, 1755113, 140404442, 19400886875, 4764856837927, 2050537030592506, 1572046460892726633, 2133798146501117397613, 5167591018292995062973870, 22288638410038000574365307819, 171813750317653145879779979300275, 2366759768251260378737273078723819964

Offset: 0

R. H. Hardin, Sep 04 2008

nonn

			Configurations of 0,1,2,3 or 4 kings on an 2 X 2 chessboard and the number of mutual attacks:
.. .. .. K. .K .. K. KK .K .K K. K. .K KK KK KK
.. K. .K .. .. KK K. .. .K K. .K KK KK .K K. KK
0  0  0  0  0  1  1  1  1  1  1  3  3  3  3  6
From 16 configurations is 11 with no more than 1 mutual attack, a(2)=11.

R. H. Hardin, Table of n, a(n) for n = 0..26

Cf. A143875, A143881, A143886.
Cf. A143871, A143872, A143873, A143874.
Diagonal of A268781.

Example added by Vaclav Kotesovec, Oct 20 2014

Additional terms from R. H. Hardin, Feb 13 2016

A268775 Number of n X 2 binary arrays with some element plus some horizontally, vertically, diagonally or antidiagonally adjacent neighbor totalling two no more than once.

Original entry on oeis.org

4, 11, 26, 65, 148, 343, 766, 1709, 3752, 8195, 17746, 38233, 81916, 174767, 371366, 786437, 1660240, 3495259, 7340026, 15379121, 32156324, 67108871, 139810126, 290805085, 603979768, 1252698803, 2594876066, 5368709129, 11095332172

Offset: 1

R. H. Hardin, Feb 13 2016

nonn

			Some solutions for n=4:
..0..0. .0..0. .0..1. .0..1. .1..0. .0..0. .0..0. .1..0. .1..1. .0..0
..0..0. .1..1. .1..0. .0..0. .0..0. .1..1. .0..0. .0..0. .0..0. .0..1
..0..1. .0..0. .0..0. .1..1. .0..0. .0..0. .1..0. .0..1. .0..1. .0..0
..1..0. .1..0. .1..0. .0..0. .1..0. .0..1. .1..0. .1..0. .0..0. .1..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 2 of A268781.

Empirical: a(n) = 2*a(n-1) + 3*a(n-2) - 4*a(n-3) - 4*a(n-4).
Conjectures from Colin Barker, Jan 15 2019: (Start)
G.f.: x*(4 + 3*x - 8*x^2 - 4*x^3) / ((1 + x)^2*(1 - 2*x)^2).
a(n) = ((-1)^(1+n) + 2^(2+n) + ((-1)^n+2^(1+n))*n) / 3.
(End)

A268776 Number of n X 3 binary arrays with some element plus some horizontally, vertically, diagonally or antidiagonally adjacent neighbor totalling two no more than once.

Original entry on oeis.org

7, 26, 91, 316, 1031, 3354, 10615, 33344, 103339, 317958, 970515, 2945172, 8888719, 26705714, 79909167, 238257768, 708129267, 2098664158, 6203795403, 18296271036, 53845375703, 158159174410, 463734769895, 1357486034320, 3967761581627

Offset: 1

R. H. Hardin, Feb 13 2016

nonn

			Some solutions for n=4:
..0..1..0. .0..0..0. .0..0..0. .0..0..0. .0..0..0. .0..0..0. .0..1..0
..0..1..0. .0..1..1. .1..0..0. .0..0..0. .0..0..1. .0..1..0. .0..1..0
..0..0..0. .0..0..0. .0..0..1. .0..0..1. .1..0..0. .0..0..0. .0..0..0
..0..1..0. .0..1..0. .0..1..0. .1..0..0. .1..0..0. .0..0..1. .1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 3 of A268781.

Empirical: a(n) = 4*a(n-1) + 2*a(n-2) - 16*a(n-3) - a(n-4) + 12*a(n-5) - 4*a(n-6).

Empirical g.f.: x*(7 - 2*x - 27*x^2 + 12*x^3 + 8*x^4 - 4*x^5) / (1 - 2*x - 3*x^2 + 2*x^3)^2. - Colin Barker, Jan 15 2019

A268777 Number of n X 4 binary arrays with some element plus some horizontally, vertically, diagonally or antidiagonally adjacent neighbor totalling two no more than once.

Original entry on oeis.org

13, 65, 316, 1462, 6383, 27531, 115391, 478849, 1957904, 7940136, 31916445, 127480373, 506131101, 1999695453, 7865869056, 30823236470, 120372357259, 468663337303, 1819741296607, 7048393305965, 27239539562644, 105056982554032

Offset: 1

R. H. Hardin, Feb 13 2016

nonn

			Some solutions for n=4:
..0..0..0..0. .1..0..0..0. .0..1..0..0. .1..0..1..0. .0..0..0..1
..1..0..1..0. .0..0..0..1. .1..0..0..0. .0..0..0..0. .1..0..0..0
..0..0..1..0. .1..0..0..0. .0..0..0..0. .0..0..1..0. .1..0..0..0
..1..0..0..0. .1..0..1..0. .1..0..0..0. .1..0..0..1. .0..0..0..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 4 of A268781.

Empirical: a(n) = 4*a(n-1) + 10*a(n-2) - 32*a(n-3) - 47*a(n-4) + 40*a(n-5) + 38*a(n-6) - 12*a(n-7) - 9*a(n-8).

Empirical g.f.: x*(13 + 13*x - 74*x^2 - 36*x^3 + 66*x^4 + 26*x^5 - 21*x^6 - 9*x^7) / (1 - 2*x - 7*x^2 + 2*x^3 + 3*x^4)^2. - Colin Barker, Jan 15 2019

A268778 Number of nX5 binary arrays with some element plus some horizontally, vertically, diagonally or antidiagonally adjacent neighbor totalling two no more than once.

Original entry on oeis.org

23, 148, 1031, 6383, 38483, 224960, 1288693, 7271509, 40511381, 223527424, 1223021243, 6646278717, 35903716877, 192977652868, 1032630435883, 5504232072227, 29238119105311, 154834459041646, 817685821067843

Offset: 1

R. H. Hardin, Feb 13 2016

nonn

Column 5 of A268781.

			Some solutions for n=4
..1..1..0..1..0. .1..1..0..0..0. .1..1..0..0..1. .0..1..0..0..0
..0..0..0..0..0. .0..0..0..0..1. .0..0..0..0..0. .0..0..0..1..1
..0..0..0..0..0. .0..0..0..0..0. .0..0..0..0..0. .0..0..0..0..0
..0..0..1..0..0. .1..0..1..0..0. .0..1..0..0..0. .0..0..0..0..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A268781.

Empirical: a(n) = 4*a(n-1) +28*a(n-2) -62*a(n-3) -314*a(n-4) +78*a(n-5) +867*a(n-6) +6*a(n-7) -859*a(n-8) +46*a(n-9) +215*a(n-10) -8*a(n-11) -16*a(n-12)

A268779 Number of nX6 binary arrays with some element plus some horizontally, vertically, diagonally or antidiagonally adjacent neighbor totalling two no more than once.

Original entry on oeis.org

41, 343, 3354, 27531, 224960, 1755113, 13493468, 101738555, 758303322, 5590121407, 40870469356, 296640792103, 2140108184248, 15358691305417, 109723986174308, 780748875032869, 5535897115345958, 39128843941494495

Offset: 1

R. H. Hardin, Feb 13 2016

nonn

Column 6 of A268781.

			Some solutions for n=4
..0..0..0..0..0..0. .0..0..0..1..1..0. .0..0..0..0..0..0. .0..0..0..0..0..0
..0..0..1..0..0..1. .1..0..0..0..0..0. .0..1..0..1..0..1. .0..0..0..1..0..0
..1..0..0..0..0..0. .0..0..0..0..0..0. .0..0..0..0..0..0. .0..0..0..0..0..1
..0..0..1..0..1..0. .1..0..0..0..1..0. .1..0..0..0..1..0. .1..0..1..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A268781.

Empirical: a(n) = 6*a(n-1) +51*a(n-2) -214*a(n-3) -1074*a(n-4) +2018*a(n-5) +7713*a(n-6) -10572*a(n-7) -22926*a(n-8) +30116*a(n-9) +25283*a(n-10) -32400*a(n-11) -15148*a(n-12) +15184*a(n-13) +5660*a(n-14) -2688*a(n-15) -1024*a(n-16)

A268780 Number of n X 7 binary arrays with some element plus some horizontally, vertically, diagonally or antidiagonally adjacent neighbor totalling two no more than once.

Original entry on oeis.org

72, 766, 10615, 115391, 1288693, 13493468, 140404442, 1425678976, 14341399141, 142487073304, 1404716302427, 13742060955231, 133640514636584, 1292631496259982, 12446235637750465, 119353876258739189

Offset: 1

R. H. Hardin, Feb 13 2016

nonn

Column 7 of A268781.

			Some solutions for n=4
..0..0..0..0..1..0..0. .0..1..0..0..0..0..0. .1..0..1..0..0..1..0
..0..0..0..0..0..0..0. .0..0..0..1..0..1..0. .0..0..0..0..0..0..1
..0..0..0..0..1..0..0. .0..0..0..0..0..0..0. .1..0..0..0..0..0..0
..0..0..0..1..0..0..0. .1..0..0..0..0..0..1. .0..0..0..0..0..0..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A268781.

Empirical: a(n) = 12*a(n-1) +64*a(n-2) -942*a(n-3) -1476*a(n-4) +26868*a(n-5) +2249*a(n-6) -376788*a(n-7) +333472*a(n-8) +2686292*a(n-9) -4376424*a(n-10) -8985248*a(n-11) +21881197*a(n-12) +12658940*a(n-13) -55768960*a(n-14) +923990*a(n-15) +80699088*a(n-16) -25850884*a(n-17) -69171189*a(n-18) +34934900*a(n-19) +34833816*a(n-20) -22502076*a(n-21) -9502460*a(n-22) +7752808*a(n-23) +1023260*a(n-24) -1356480*a(n-25) +55136*a(n-26) +94080*a(n-27) -14400*a(n-28).

cp's OEIS Frontend

A143870 Number of ways of placing kings with no more than 1 mutual attack on an n X n chessboard.

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A268775 Number of n X 2 binary arrays with some element plus some horizontally, vertically, diagonally or antidiagonally adjacent neighbor totalling two no more than once.

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A268776 Number of n X 3 binary arrays with some element plus some horizontally, vertically, diagonally or antidiagonally adjacent neighbor totalling two no more than once.

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A268777 Number of n X 4 binary arrays with some element plus some horizontally, vertically, diagonally or antidiagonally adjacent neighbor totalling two no more than once.

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A268778 Number of nX5 binary arrays with some element plus some horizontally, vertically, diagonally or antidiagonally adjacent neighbor totalling two no more than once.

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A268779 Number of nX6 binary arrays with some element plus some horizontally, vertically, diagonally or antidiagonally adjacent neighbor totalling two no more than once.

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A268780 Number of n X 7 binary arrays with some element plus some horizontally, vertically, diagonally or antidiagonally adjacent neighbor totalling two no more than once.

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